#2228212

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2864 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

(6,9)(10,12)(13,14)(15,19)(16,20)(17,18)(21,22,24,26)(23,27,28,25);(1,2,3,4)(5,6,8,9)(7,10,11,12)(13,15)(14,16)(17,20)(18,19)(21,22)(23,25)(24,26)(27,28);(1,2)(3,4)(7,11)(10,12)(14,18)(16,19)(21,23,24,28)(22,25,26,27);(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,20)(18,19);(5,7)(6,10)(8,11)(9,12)(13,16)(14,15)(17,19)(18,20)(21,22)(23,27)(24,26)(25,28)

Polycyclic presentation:

2^9|a^2=fi,b^2=fgh,c^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2135408);